Find Value of m for Total Valid Relations

To find the value of $m$, we first need to determine the total number of ordered pairs $(a, b) \in S \times S$ that satisfy the condition $|a - b| \ge 2$, where $S = \{1, 2, 3, 4, 5, 6\}$.

Let $P$ be the set of all such ordered pairs: $P = \{(a, b) \in S \times S : |a - b| \ge 2\}$

The total number of elements in $S \times S$ is: $n(S \times S) = 6 \times 6 = 36$

Now, we count the number of ordered pairs $(a, b)$ that do not satisfy $|a - b| \ge 2$, which means $|a - b| < 2$ (i.e., $|a - b| = 0$ or $|a - b| = 1$):

1. Case 1: $|a - b| = 0$ The possible pairs are $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)$. Number of pairs = $6$

2. Case 2: $|a - b| = 1$ The possible pairs are $(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), (5, 6), (6, 5)$. Number of pairs = $10$

Thus, the number of pairs with $|a - b| < 2$ is: $6 + 10 = 16$

Therefore, the number of valid pairs in $P$ is: $n(P) = n(S \times S) - 16 = 36 - 16 = 20$

A relation $R \in X$ is a subset of $P$ containing exactly $6$ elements. The total number of such relations is given by choosing $6$ pairs from the $20$ available pairs in $P$: $n(X) = {}^{20}C_6$

Comparing this with the given expression $n(X) = {}^m C_6$, we get: $m = 20$